Could an organism evolve to adaptively evolve? Besides chance mutations sticking around because of their utility towards survival and procreation, could an organism evolve the ability to aid this process by mutating more frequently in beneficial directions in response to the environment (e.g. based on some form of past higher order evolutionary experience)?

IMHO, the whole genome replication/recombination/mutation process and arrangement has evolved to increase the likelihood that changes will be beneficial or at least neutral. In chromosome recombination, for example, new chromosome combinations are constructed from old, tested, gene sequences, while making sure that the new pieces are homologous to corresponding pieces in the parents' and grandparents' chromosomes.